L31_Non-Linear Programming Problems - Unconstrained Optimization - KKT Conditions

L31_Non-Linear Programming Problems - Unconstrained Optimization - KKT Conditions

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Unformatted text preview: CLASSICAL OPTIMIZATION THEORY Quadratic forms Let = n x x x X . . 2 1 be a n-vector. Let A = ( a ij ) be a nn symmetric matrix. We define the k th order principal minor as the k k determinant kk k k k k a a a a a a a a a . .. . . . .. . .. 2 1 2 22 21 1 12 11 Positive semi definite if X T AX 0 for all X. Positive definite if X T AX > 0 for all X 0. Negative semi definite if X T AX 0 for all X . Negative definite if X T AX < 0 for all X 0. Then the quadratic form 2 1 2 ii i ij i j i i j n a x a x x < = + X A X X Q T = ) ( (or the matrix A) is called A necessary and sufficient condition for A (or Q ( X )) to be : Negative definite ( negative semi definite ) if k th principal minor of A has the sign of (-1) k , k =1,2,,n ( k th principal minor of A is zero or has the sign of (-1) k , k =1,2,,n ) Positive definite ( positive semi definite ) is that all the n principal minors of A are > 0 ( 0). Let f ( X )= f ( x 1 , x 2 ,, x n ) be a real-valued function of the n variables x 1 , x 2 ,, x n (we assume that f ( X ) is at least twice differentiable ). 0 0 ( ) ( ) f X h f X + r j h ) ,.., , ( 2 1 n h h h h = r 0 0 0 0 1 1 2 2 ( , ,.., ) n n X h x h x h x h + = + + + r A point X is said to be a local maximum of f ( X ) if there exists an > 0 such that for all Here and 1 2 ( , ,.., ) n X x x x = X is said to be a local minimum of f ( X ) if there exists an > 0 such that X X f X f 2200 ) ( ) ( ) ( ) ( X f h X f + r X is called an absolute maximum or global maximum of f ( X ) if for all X is called an absolute minimum or global minimum of f ( X ) if ( ) ( ) f X f X X 2200 j h Theorem A necessary condition for X to be an optimum point of f ( X ) is that ) ( = X f i x f (that is all the first order partial derivatives are zero at X .) ) ( = X f Definition is called a stationary point of f ( X ) (potential candidate for local maximum or local minimum). A point X for which Let X be a stationary point of f ( X ). A sufficient condition for X to be a local minimum of f ( X ) is that the Hessian matrix H ( X ) is positive definite; local maximum of f ( X ) is that the Hessian matrix H ( X ) is negative definite . Theorem Here H ( X ) is the nn matrix whose i th row are j f x 2 2 2 2 1 1 2 1 2 2 2 2 1 2 2 2 2 2 2 2 1 2 . . . . . . . . . . n n n n n f f f x x x x x f f f x x x x x f f f x x x x x i.e. H ( X ) = with respect to x i ....
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L31_Non-Linear Programming Problems - Unconstrained Optimization - KKT Conditions

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